The belief that there are indivisible units of extension is termed 'atomism'. The opposing view is that extension is infinitely divisible. For the purpose of this work, I shall occasionally refer to this position as 'divisionism' and the adherents to this view as 'divisionists'.
Preliminary studies showed that some of the traditional arguments supporting infinite divisibility make use of premisses which effectively beg the question. The same appears true of Atomism. In this work I show that the traditional mathematical arguments for infinite divisibility are flawed and that most philosophers in the past did not discover the flaw. My view is that the two positions, atomism and divisionism, are each internally consistent and, though mutually incompatible, are independent in a way not unlike Euclidean and non-Euclidean geometries or the waves and particles of quantum physics.